I skip my proof of this theorem because it is exactly analogous to Potato's proof; with two modifications: (a) our vectors are in $\mathbb{R}^d$, rather than $\mathbb{C}$ or $\mathbb{R}^2$); (b) I am using the notation $v_i$ in place of $z_i$ here, since it feels more natural in this context.

Now, my question (slightly informally stated) is

Question. For any fixed $d$, what is the largest value $c_d^{\ast}$ of $c_d$ for which the above theorem is true?

If we cannot find the exact value $c_d^{\ast}$, it will be interesting to find nontrivial upper and lower bounds. Working out the details of the above proof, we can show that $c_d$ can be taken to be (at least) $\frac{1}{2d}$; i.e., $c_d^{\ast} \geqslant \frac{1}{2d}$. Can we say anything better?

It is easy to see that $c_1 = \frac{1}{2}$, which matches the above bound for $d=1$. Robert Israel's answer shows that $c_2 = \frac{1}{\pi}$ (exactly!), which is strictly better than $\frac{1}{2 \cdot 2} = \frac{1}{4}$. However, his proof seems to make an essential use of complex numbers that I could not generalise to $d$ dimensions.

1 Answer
1

Define
$$
R(x)=\left\{\begin{array}{ll}x_1&\text{when }x_1\ge0\\0&\text{when }x_1\lt0\end{array}\right.\tag{1}
$$
Integrating $R(x)$ over the sphere of radius $r$ yields
$$
\omega_{d-2}\;r^d\int_0^{\pi/2}\cos^{d-2}(\theta)\sin(\theta)\;\mathrm{d}\theta=\omega_{d-2}\;\frac{r^d}{d-1}\tag{2}
$$
Integrating $1$ over the sphere of radius $r$ yields
$$
\begin{align}
\omega_{d-2}\;r^{d-1}\int_{-\pi/2}^{\pi/2}\cos^{d-2}(\theta)\;\mathrm{d}\theta
&=2\;\omega_{d-2}\;r^{d-1}\int_0^{\pi/2}\cos^{d-2}(\theta)\;\mathrm{d}\theta\\
&=\omega_{d-2}\;r^{d-1}\mathrm{B}\left(\frac{d-1}{2},\frac12\right)\\
&=\omega_{d-2}\;r^{d-1}\frac{\Gamma\left(\frac{d-1}{2}\right)\Gamma\left(\frac12\right)}{\Gamma\left(\frac d2\right)}\tag{3}
\end{align}
$$
using the identity
$$
\int_0^{\pi/2}\sin^{\alpha-1}(t)\;\cos^{\beta-1}(t)\;\mathrm{d}t
=\frac12\mathrm{B}\left(\frac{\alpha}{2},\frac{\beta}{2}\right)\tag{4}
$$
Therefore, the mean of $R(x)$ over the surface of the sphere of radius $r$ is $(2)$ divided by $(3)$:
$$
\frac{r\;\Gamma\left(\frac{d}{2}\right)}{(d-1)\Gamma\left(\frac{d-1}{2}\right)\Gamma\left(\frac{1}{2}\right)}=\frac{r\;\Gamma\left(\frac{d}{2}\right)}{2\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{d+1}{2}\right)}\tag{5}
$$
Thus, the mean of $\displaystyle\sum_{k=1}^nR(x_j)$ over all rotations of the sphere is
$$
\frac{\Gamma\left(\frac{d}{2}\right)}{2\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{d+1}{2}\right)}\sum_{k=1}^n|x_k|\tag{6}
$$
Thus, for some rotation of the sphere, we must have
$$
\sum_{k=1}^nR(x_k)\ge\frac{\Gamma\left(\frac{d}{2}\right)}{2\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{d+1}{2}\right)}\sum_{k=1}^n|x_k|\tag{7}
$$
To see that $(7)$ is the best possible independent of $n$, consider an "even" distribution of $n$ points as $n\to\infty$. This estimates the $(d{-}1)$-dimensional analog of the Riemann sum for the integral in $(2)$.

As shown in the appendix to this answer, $\displaystyle\lim_{x\to\infty}\frac{\Gamma(x+\alpha)}{x^\alpha\Gamma(x)}=1$. Therefore, asymptotically,
$$
c_d\sim\dfrac{1}{\sqrt{2\pi d}}\tag{9}
$$
which is the geometric mean of $\dfrac{1}{\pi}$ and $\dfrac{1}{2d}$, the upper bound derived in the question.